How many permutations can be formed from the word “VOWELS” so that (i) There is no restriction on letters (ii) Each word begins with E (iii) Each word begins with O and ends with L (iv) All vowels are together (v) All consonants are together (vi) All vowels are never together

(i) The word VOWELS consists of 6 distinct letters that can be arranged amongst themselves in 6! ways.
∴ Number of words that can be formed with the letters of the word VOWELS, without any restriction = 6! = 720

(ii) If we fix the first letter as E, the remaining 5 letters can be arranged in 5! ways to form the words.
∴ Number of words starting with the E = 5! = 120

(iii) If we fix the first letter as O and the last letter as L, the remaining 4 letters can be arranged in 4! ways to form the words.
∴ Number of words that start with O and end with L = 4! = 24

(iv) The word VOWELS consists of 2 vowels.
If we keep all the vowels together, we have to consider them as a single entity.
Now, we are left with the 4 consonants and all the vowels that are taken together as a single entity.
This gives us a total of 5 entities that can be arranged in 5! ways.
It is also to be considered that the 2 vowels can be arranged in 2! ways amongst themselves.

By fundamental principle of counting:
∴ Total number of arrangements = 5!×2! = 240

(v) The word VOWELS consists of 4 consonants.
If we keep all the consonants together, we have to consider them as a single entity.
Now, we are left with the 2 vowels and all the consonants that are taken together as a single entity.
This gives us a total of 3 entities that can be arranged in 3! ways.
It is also to be considered that the 4 consonants can be arranged in 4! ways amongst themselves.

By fundamental principle of counting:
∴ Total number of arrangements = 3!×4! = 144
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